Reflexivity of function spaces associated to a σ-finite vector measure

Reflexi i y o unc ion spaces associa ed o a σ-fini e ec o measu e ✩ Rica do del Campo a, An onio Fe nández b,∗, Fe nando Mayo al b, F ancisco Na anjo b a Dp o. Ma emá ica Aplicada I, Uni e sidad de Se illa, EUITA, C a. de U e a Km. 1, 41013 Se illa, Spain b Dp o. Ma emá ica Aplicada II, Escuela Técnica Supe io de Ingenie ía, Camino de los Descub imien os, s/n, 41092 Se illa, Spain a b s a c Keywo ds: Reflexi i y In eg able unc ion Vec o measu e δ- ing Locally s ongly addi i e measu e Fo a ec o measu e νdefined on a δ- ing wi h alues in a Banach space and 1 < p <∞, we cha ac e ize he eflexi i y o he diffe en spaces Lp w(ν) (in eg abili y in he weak sense), Lp(ν) (in eg abili y in he s ong sense), and Lp(ν) (in eg abili y in he Choque sense). 1. In oduc ion F om he poin o iew o unc ional analysis he second mos desi ed p ope y o infini e spaces is eflexi i y ( he fi s one is comple eness) and p obably i is he mos used in applica ions due o he weak compac ness o i s uni ball. Typical unde g adua e examples o eflexi e Banach spaces a e Lebesgue Lp-spaces (1 <p <∞) o a posi i e σ-fini e measu e. The co esponding scala unc ion spaces associa ed o a ec o measu e νwi h alues in o a Banach space ha e been long s udied (see, o example [18] and mos o he e e ences in he p esen pape ). In his new con ex he hings a e eally diffe en . The e appea se e al Lp-spaces associa ed o he ec o measu e: in he weak sense Lp w(ν), in he s ong sense Lp(ν), and finally, in eg abili y in he Choque sense Lp(ν), o cou se o 1 ≤p <∞. These kind o spaces a e, in gene al, diffe en om each o he and non eflexi e, e en o 1 <p <∞. When he ec o measu e νis ✩This esea ch has been pa ially suppo ed by La Jun a de Andalucía. The au ho s acknowledge he suppo o he Minis e io de Economía y Compe i i idad o Spain and FEDER, unde he p ojec MTM2012-36740-C02-01. *Co esponding au ho . E-mail add esses: camp[email p o ec ed] (R. del Campo), [email p o ec ed] (A. Fe nández), may[email p o ec ed] (F. Mayo al), [email p o ec ed] (F. Na anjo). . defined on a σ-algeb a he eflexi i y o Lp w(ν)and Lp(ν)has been s udied in [12]. Roughly speaking, o 1 <p <∞, he space Lp w(ν), o equi alen ly Lp(ν), is eflexi e i and only i hey coincide. Also in he same con ex o a ec o measu e defined on a σ-algeb a, he eflexi i y o Lp(ν)is ob ained as a byp oduc o a gene al esul abou in e pola ion om [10], namely, Lp(ν)is always eflexi e o all 1 <p <∞. In he p esen pape we s udy he eflexi i y o hese spaces when he measu e is defined on a δ- ing, a mo e gene al (bu na u al) s uc u e han a σ-algeb a. In his new con ex we can say ha a simila esul cha ac e izing eflexi i y o Lp w(ν)and Lp(ν)holds (see Theo em 2.3). Ne e heless Lp(ν)is no always eflexi e. We cha ac e ize hose ec o measu es o which Lp(ν)is eflexi e as he locally s ongly addi i e ec o measu es (see Theo em 4.3). Much o his wo k deals wi h his kind o measu es. 2. Reflexi i y o Lpand Lp w The basic e e ences o us abou in eg a ion will be [7,13,16,17] and [18, Chap e 3]. Th oughou his pape we will conside a ec o measu e ν:R →Xdefined on a δ- ing Ro subse s o some nonemp y se Ωwi h alues in a eal Banach space X, wi h dual X. We deno e by Rloc he σ-algeb a o subse s A ⊆Ω such ha A ∩B∈R o each B∈R. Measu abili y o unc ions :Ω −→ Rwill be conside ed wi h espec o he measu able space (Ω, Rloc). The semi a ia ion o νis he se unc ion ν :Rloc →[0, ∞] defined by ν(A) := sup {|ν, x| (A):xX≤1}, whe e |ν, x| is he a ia ion o he scala measu e ν, x:A∈R−→  ν, x(A):=ν(A),x ∈R. Recall ha o e e y subse A ∈Rloc, we ha e he ollowing inequali ies 1 2ν(A)≤sup{ν(B):B∈R,B ⊆A}≤ν(A). The semi a ia ion is a subaddi i e se unc ion ha may be nonaddi i e. A se N∈Rloc is called ν-null i ν(N) =0, and a p ope y holds ν-almos e e ywhe e (ν-a.e.) i i holds excep on a ν-null se . In wha ollows we will always conside ec o measu es ν:R →Xwhich a e σ-fini e, ha is, he e exis a pai wise disjoin sequence (Ωk)kin R, and a ν-null se N∈Rloc, such ha Ω =(∪k≥1Ωk)∪N. Simple examples o σ-fini e ec o measu es defined on δ- ings a e gi en by he Lebesgue measu e λdefined on he δ- ing R := {A∈M:λ(A)<∞}, whe e Mis he σ-algeb a o all Lebesgue measu able subse s o he eal line R, and he coun ing measu e defined on he δ- ing P (N)o fini e subse s o he na u al numbe s N. O he examples o σ-fini e ec o measu e will be conside ed in Examples 3.2 and 4.5 below. Mo eo e , σ-fini e ec o measu es ha e special scala con ol measu es as we see in he ollowing esul (see [7, Theo em 3.3]). Lemma 2.1. Le νbe a σ-fini e ec o measu e. Then he e exis s x 0∈X, wi h x 0X≤1, such ha |ν, x 0|(A) =0i and only i ν(A) =0, wi h A ∈Rloc. P oo . I νis σ-fini e, hen he e exis s 0 < ∈L1(ν). Conside he ec o measu e ν :Rloc →Xdefined by ν (A) := A dν ∈X. No e ha ν is defined on a σ-algeb a, and ν (A) = χ AL1(ν), o all A ∈Rloc (see [13, Theo em 3.2]). Le x 0∈X, wi h x 0X≤1, such ha |ν , x 0| is a Rybako con ol measu e o ν (see [9, Theo em IX.1.2]). Then |ν, x 0|(A) =0i and only i ν(A) =0, wi h A ∈Rloc, because we know ha |ν , x 0|(A) =A d|ν, x 0|, o all A ∈Rloc.2 A measu able unc ion :Ω −→ Ris called weakly in eg able (wi h espec o ν) i ∈L1(|ν, x|) o all x∈X. A weakly in eg able unc ion is said o be in eg able (wi h espec o ν) i , o each A ∈Rloc he e exis s an elemen (necessa ily unique) A dν ∈X, sa is ying  A dν,x= A dν, x,x ∈X. I 1 ≤p <∞, a measu able unc ion :Ω −→ Ris called weakly p-in eg able (wi h espec o ν) i | |p is weakly in eg able and p-in eg able (wi h espec o ν) i | |pis in eg able. The space Lp w(ν)o all (ν-a.e. equi alence classes o ) weakly p-in eg able unc ions becomes a Banach la ice when endowed wi h he usual ν-a.e. poin wise o de and he no m  Lp w(ν):= sup ⎧ ⎪ ⎨ ⎪ ⎩⎛ ⎝ Ω | |pd|ν, x|⎞ ⎠ 1 p :xX≤1⎫ ⎪ ⎬ ⎪ ⎭ . Mo eo e , he space Lp(ν)o all (ν-a.e. equi alence classes o ) p-in eg able unc ions is a closed o de con inuous ideal o Lp w(ν). In ac , i is he closu e o S(R), he space o simple unc ions suppo ed on R(see [13, Theo em 3.5]). Recall ha o de con inuous means ha  − nLp(ν)→0 o e e y 0 ≤ n↑ ∈Lp(ν). Fo p ≥1, no e ha Lp w(ν)={ :Ω−→ R:| |p∈L1 w(ν)}, Lp w(ν)=| |p 1 p L1 w(ν). These Banach la ices Lp(ν)and Lp w(ν)we e ini ially s udied in [12] and [19] o ec o measu es νdefined on a σ-algeb a and i s basic p ope ies can be ex ended and emain ue o ec o measu es defined on δ- ings (see [4]). Le us men ion, in pa icula , ha Lp w(ν)is p-con ex, ha is, he e is a cons an K>0 such ha   (| 1|p+···+| n|p)1 p  Lp w(ν)≤K 1p Lp w(ν)+···+ np Lp w(ν)1 p, o e e y elec ion o ec o s 1, ..., nin Lp w(ν), as we can see di ec ly om he defini ion o he no m  · Lp w(ν). The ollowing esul has been bo owed om [4, p. 75] (see also [2, Co olla y 5.7]). We include he e he p oo o he sake o comple eness. P oposi ion 2.2. Le 1 ≤p <∞and le 0 ≤ n↑in Lp w(ν)such ha supn nLp w(ν)<∞. Then, he e exis s supn n∈Lp w(ν). Mo eo e supn nLp w(ν)=supn nLp w(ν). Tha is, Lp w(ν)has he sequen ial Fa ou p ope y. P oo . The e exis s a ν-null se N∈Rloc such ha 0 ≤ n(w) ↑ o all w∈Ω N. Conside he unc ion g:Ω −→ [0, ∞] defined by g(w) := supn n(w), i w∈Ω Nand g(w) =0, i w∈N. Then we ha e 0 ≤ p nχΩN↑gppoin wise, and he Lebesgue mono one con e gence heo em assu es ha  Ω gpd|ν, x| = lim n Ω p nχΩNd|ν, x| ≤ xsup n  np Lp w(ν)<∞, o all x∈X. In his way g∈Lp(|ν, x|) o all x∈X, and sup ⎧ ⎨ ⎩ Ω gpd|ν, x| :x≤1⎫ ⎬ ⎭ ≤sup n  np Lp w(ν)<∞. In pa icula , by applying he abo e o he ec o x 0o Lemma 2.1, we deduce ha gis fini e ν-a.e. and, in ac , i equals wi h supn n. Thus g=sup n n∈Lp w(ν), and mo eo e     sup n n   Lp w(ν) =gLp w(ν)≤sup n  nLp w(ν)≤    sup n n   Lp w(ν) .2 Recall ha a Banach la ice is a KB-space whene e e e y no m bounded, posi i e, inc easing sequence is no m con e gen [1, Defini ion 14.10]. Thus e e y eflexi e space is a KB-space (see he commen s o he a o emen ioned defini ion), and i is clea ha e e y KB-space has o de con inuous no m. Mo eo e e e y KB-space has he sequen ial Fa ou p ope y because e e y con e gen (in no m) inc easing sequence, neces- sa ily con e ges o i s sup emum. The nex esul is he analogue o [12, Co olla y 3.10] o ec o measu es defined on δ- ings. I s p oo is a small modifica ion o ha , bu we include i he e o he sake o comple e- ness. The equi alence o d)and h)has been p o ed independen ly by A alos-Ramos and Galaz-Fon es in [2, Co olla y 5.20]. Theo em 2.3. Fo e e y p >1, he ollowing condi ions a e equi alen : a) Lp w(ν)has o de con inuous no m. b) Lp w(ν)is a KB-space. c) Lp w(ν)is eflexi e. d) Lp(ν)is eflexi e. e) Lp(ν)is a KB-space. ) Lp(ν)has he sequen ial Fa ou p ope y. g) Lp w(ν) =Lp(ν)as Banach la ices. h) L1 w(ν) =L1(ν)as Banach la ices. All eigh asse ions a e ue whene e he Banach space Xis weakly sequen ially comple e. P oo . a) =⇒b)Le ( n)nbe a no m bounded, posi i e, inc easing sequence in Lp w(ν). By applying P opo- si ion 2.2, he e exis s in Lp w(ν)such ha n↑ . Then, om o de con inui y o he no m, we ha e ha ( n)ncon e ges o in Lp w(m). b) =⇒c)S ince Lp w(ν)is a p-con ex (wi h p >1) Banach la ice, he space o summable sequences 1is no la ice embeddable in Lp w(ν)(see [14, p. 51]). Mo eo e , Lp w(ν)does no con ain a la ice copy o he space o null sequences c0since i is a KB-space by hypo hesis (see [1, Theo em 14.12]). The esul hen ollows om Lozano skii’s esul (see [1, Theo em 14.23]). c) =⇒d)Lp(ν)is a closed subspace o Lp w(ν). d) =⇒e)I is well known ha eflexi e spaces a e KB-spaces. e) =⇒ )E e y KB-space has he sequen ial Fa ou p ope y. ) =⇒g)S ee [4, P oposi ion 5.4]. g)⇐⇒ h)I is enough o obse e ha ∈L1 w(ν)i and only i | |1 p∈Lp w(ν). g) =⇒a)No e ha Lp(ν)has always o de con inuous no m. See [19, P oposi ion 6] o [13, Theo em 3.3]. Fo he las claim in he s a emen o he heo em, ecall ha L1 w(ν) =L1(ν) whene e he Banach space Xis weakly sequen ially comple e. See [13, Theo em 5.1].2 3. Fa ou p ope y and o de con inui y o Lpo he semi a ia ion Now we a e going o conside , o 1 ≤p <∞, he spaces deno ed by Lp(ν). These spaces appea in a na u al way, as Lo en z spaces wi h espec o he semi a ia ion ν, when we desc ibe he in e pola ion spaces ob ained by applying he eal in e pola ion me hod o couples o Lp-spaces o a ec o measu e ν:R →X(see [6] and [10]). Le us in oduce i b iefly and desc ibe some basic p ope ies o hem. Gi en a measu able unc ion :Ω −→ R, we shall conside i s dis ibu ion unc ion (wi h espec o he semi a ia ion o he ec o measu e ν) ν : ∈[0, ∞) −→  ν ( ) ∈[0, ∞], defined by ν ( ):=ν({w∈Ω:| (w)|> }), ≥0. This dis ibu ion unc ion has simila p ope ies as in he scala case (see [10]). Fo ins ance, ν is non-inc easing and igh -con inuous. Recall ha L1(ν)is he space o (ν-a.e. equi alence classes o ) measu able unc ions :Ω −→ Rsuch ha he in eg al ∞ 0ν ( )d <∞. Then L1(ν), wi h he quasi-no m  L1(ν):= ∞ 0ν ( )d and he usual ν-a.e. poin wise o de , becomes a quasi-Banach la ice. Fo 1 <p <∞, we also conside he space Lp(ν):= :Ω−→ R:| |p∈L1(ν), wi h he quasi-no m  Lp(ν):= | |p 1 p L1(ν). We would need o men ion ha a consequence o [6, Rema k 3.8.1] is ha Lp(ν)is no mable o e e y 1 <p <∞. This means ha he e is a la ice no m  · pequi alen o he quasi-no m  · Lp(ν). The case p =1is some hing special because we don’ know i L1(ν)is no mable (see [11] o de ails). The ollowing esul is he analogue o P oposi ion 2.2. P oposi ion 3.1. Le 1 ≤p <∞and le 0 ≤ n↑in Lp(ν)such ha supn nLp(ν)<∞. Then, he e exis s supn n∈Lp(ν). Mo eo e supn nLp(ν)=supn nLp(ν). Tha is, Lp(ν)has he sequen ial Fa ou p ope y. P oo . The e exis s a subse N∈Rloc, wi h ν(N) =0, such ha 0 ≤ n(w) ↑ o all w∈Ω N. Conside he unc ion g:Ω −→ [0, ∞] defined by g(w) := supn n(w), i w∈Ω Nand g(w) =0, i w∈N. Then we ha e 0 ≤ p nχΩN↑gppoin wise, and ν p nχΩN( )↑νgp( ) o all ≥0. By applying he Lebesgue mono one con e gence heo em we ob ain ∞  0 νgp( )d = lim n ∞  0 ν p nχΩN( )d = lim n ∞  0 ν p n( )d =sup n  np Lp(ν)<∞. Then νgp( ) <∞ o all >0and gis fini e ν-a.e. We conclude ha supn n∈Lp(ν)and mo eo e supn nLp(ν)=supn nLp(ν).2 As i has been poin ed ou in [10], in gene al, he spaces Lp(ν), Lp(ν)and Lp w(ν)do no coincide, and he h ee spaces can be diffe en . I he measu e νis defined on a σ-algeb a, we ha e he ollowing inclusions L∞(ν) ⊆Lp(ν) ⊆Lp(ν) ⊆Lp w(ν), and all hese inclusions a e con inuous o all 1 ≤p <∞ (see [10, P oposi ion 7]). He e L∞(ν) deno es he space o (classes ν-a.e. o ) essen ially bounded measu able unc ions :Ω −→ Rwi h he essen ial sup emum no m. Howe e , i he ec o measu e νis defined on a δ- ing ins ead o a σ-algeb a, he inclusion Lp(ν) ⊆Lp(ν)is in gene al alse as he ollowing example poin s ou . Example 3.2. (See [6, Example 2.1].) Conside he σ-fini e ec o measu e ν:A∈P (N)→ν(A):=χA∈c0. Fo e e y 1 ≤p <∞, i is easy o check ha Lp w(ν) =∞, he space o bounded sequences, and Lp(ν) =c0. In wha ollows i will be in e es ing o no e ha ν(A) =1, o e e y nonemp y A ⊆N, and ν(∅) =0. This means, in pa icula , ha ν =χ[0,∞)i is an unbounded sequence, bu ν =χ[0, ∞)i ∈∞. Consequen ly, L1(ν) =∞=L1 w(ν), and L1(ν) ⊆ L1(ν). Ne e heless, he inclusion L1(ν) ⊆L1 w(ν) emains and i is con inuous o e e y ec o measu e νdefined on a δ- ing. And, mo eo e , he inclusion L1(ν) ⊆L1(ν)holds i and only i he measu e νis locally s ongly addi i e (see [6, P oposi ion 3.2]). In pa icula , i L1(ν) =L1 w(ν), hen he measu e νis locally s ongly addi i e. Recall ha a ec o measu e νis locally s ongly addi i e i o e e y disjoin sequence (An)n⊆R, wi h ν (∪n≥1An)<∞, we ha e ν(An)X→0. See [5] and [6], whe e hese measu es we e in oduced in connec ion wi h eal and complex in e pola ion me hods and unc ion spaces associa ed o a ec o measu e. No e ha Example 3.2 ells us ha S(R), he se o simple unc ions suppo ed on subse s o he δ- ing R, is no always a dense subse o L1(ν). The hings a e diffe en i he measu e is locally s ongly addi i e. The ollowing echnical esul s will be used o p o e ha S(R)is dense in L1(ν)when he ec o measu e is locally s ongly addi i e. In wha ollows i will be con enien o conside he ollowing no a ion. Fo a measu able unc ion :Ω −→ Rand a eal numbe M, conside he measu able subse [ >M]: ={w∈Ω: (w)>M}. Simila meaning ha e [ ≤M]o [ =0]. Lemma 3.3. Le ν:R →Xbe a ec o measu e and le 0 ≤ ∈L1(ν). Then ν ([ >M]) <∞ o each M>0, and lim M→0  χ[ ≤M] L1(ν)=0. P oo . No e ha ≥Mχ [ >M], o each M>0, and so  L1(ν)≥Mχ[ >M]L1(ν)=Mν([ >M]) . Thus, ν ([ >M]) ≤1 M L1(ν)<∞. Fo he second asse ion no e ha [ χ[ ≤M]> ] =∅, i ≥M>0, and so ν [ χ[ ≤M]> ]=0 o hose . On he o he hand, i 0 ≤ <M, hen [ χ[ ≤M]> ]=[ < ≤M] and, in his case, ν [ χ[ ≤M]> ]=ν ([ < ≤M]). Thus lim M→0  χ[ ≤M] L1(ν)= lim M→0 ∞  0 ν[ χ[ ≤M]> ]d = lim M→0 M  0 ν([ < ≤M]) d ≤lim M→0 M  0 ν([ > ]) d =0, since ∈L1(ν). 2 Lemma 3.4. Le ν:R →Xbe a ec o measu e. The ollowing condi ions a e equi alen : 1) νis locally s ongly addi i e. 2) ν(En) →0 o each sequence (En)n⊆Rloc, such ha En↓∅and ν(E1) <∞. In pa icula , i νis locally s ongly addi i e, hen o e e y A ∈Rloc, wi h ν(A) <∞, and e e y ε >0 he e exis s Bε∈R, wi h Bε⊆A, such ha ν(A Bε) =χA−χBεL1(ν)<ε. P oo . 1) =⇒2) Suppose ha (En)n⊆Rloc, wi h En↓∅and ν(E1) <∞. Then χEn∈L1 w(ν) o all n ≥1 because he sequence (En)nis dec easing and ν(E1) <∞. Now, locally s ongly addi i i y o ν implies ha χEn∈L1(ν) o all n ≥1(see [6, Lemma 3.1]), and mo eo e χEn↓0poin wise in L1(ν). The o de con inui y o he no m implies ha ν(En) =χEnL1(ν)→0as we wan o see. 2) =⇒1) Le (An)n⊆Rbe a disjoin sequence wi h ν (∪n≥1An)<∞. Pu E1:= ∪n≥1Anand En:= E1(A1∪···∪An−1) o each n ≥2. Then i is clea ha (En)n⊆Rloc, En↓∅and ν(E1) <∞. Mo eo e An⊆En o all n ≥1. Thus ν(An) ≤ν(En) →0and νis locally s ongly addi i e. Fo he las asse ion ake A ∈Rloc, wi h ν(A) <∞, and ecall ha νis σ-fini e. This allows us o choose a sequence (Ωn)n⊆R, wi h Ωn↑Ω. Then A A ∩Ωn↓∅and ν(A A ∩Ω1) ≤ν(A) <∞. Now he equi alence 2) assu es ha ν(A A ∩Ωn) →0, bu ν(A A ∩Ωn) =χA−χA∩ΩnL1(ν).2 He e is he esul abou densi y o simple unc ions. P oposi ion 3.5. Le ν:R →Xbe a locally s ongly addi i e ec o measu e. Then S(R)is dense in L1(ν). P oo . Decomposing unc ions in o posi i e and nega i e pa s, i is enough o conside only nonnega i e unc ions. No e ha = χ[ >M]+ χ[ ≤M] o each 0 ≤ ∈L1(ν)and M>0. Then Lemma 3.3 assu es ha he se L1 s(ν):=g∈L1(ν):ν([g=0])<∞ is dense in L1(ν). Now we a e going o p o e ha S(Rloc) ∩L1 s(ν)is dense in L1 s(ν). Take 0 ≤ g∈L1 s(ν)and ε >0. Conside he sequence gn:= in {g, n} o all n ≥1. Then 0 ≤gn↑gand [gn=0] ⊆[g=0] o all n ≥1. Then lim n→∞ g−gnL1(ν)= lim n→∞ ∞  0 ν([g−gn> ]) d = lim n→∞ ∞  0 ν([g>n+ ]) d = lim n→∞ ∞  n ν([g>s]) ds =0. This means ha he he e exis s m ≥1such ha g−gmL1(ν)<ε 4. Since gmis bounded and [gm= 0] ⊆[g=0] he e exis s a simple unc ion ϕ := N k=1 αkχAk, wi h Ak∈Rloc, Ak⊆[g=0], αk>0, o all 1 ≤k≤Nand 0 ≤ϕ ≤gmsuch ha gm−ϕL∞(ν)<ε 4ν([g=0]) . Thus, ha ing in mind ha [gm−ϕ =0] ⊆[g=0], we ob ain gm−ϕL1(ν)= ∞  0 ν([gm−ϕ> ])d = ε 4ν([g=0])  0 ν([gm−ϕ> ])d <ε 4ν([g=0])ν([g=0])=ε 4 and, consequen ly, g−ϕL1(ν)≤2g−gmL1(ν)+2gm−ϕL1(ν)<ε. Finally, no e ha Lemma 3.4 assu es ha S(R)is dense in S(Rloc) ∩L1 s(ν). Indeed, gi en 0 ≤ϕ := n k=1 αkχAk∈S(Rloc) ∩L1 s(ν)and ε >0 he e exis s Bk∈Rsuch ha χAk−χBkL1(ν)<ε n2nn k=1 αk, o all k=1, ..., n. Now aking φ := n k=1 αkχBk∈S(R), we ob ain ha ϕ −φL1(ν)<ε, and he p oo is o e . 2 P oposi ion 3.6. Le ν:R →Xbe a ec o measu e. The ollowing condi ions a e equi alen : 1) νis locally s ongly addi i e. 2)  χEnL1(ν)→0 o e e y ∈L1(ν)and e e y sequence (En)n⊆Rloc, wi h En↓∅. 3)  − nL1(ν)→0 o e e y sequence ( n)nand o L1(ν)such ha 0 ≤ n↑ . Tha is, L1(ν) is o de con inuous. 4) Lp(ν)is o de con inuous o e e y (some) 1 ≤p <∞. P oo . 1) =⇒2) No e ha Lemma 3.4 assu es ha e e y simple unc ion ϕ ∈S(R)sa isfies he abo e condi ion 2). Gi en he unc ion ∈L1(ν), he sequence (En)n⊆Rloc, wi h En↓∅and ε >0, om P oposi ion 3.5, we know ha he e exis s ϕ ∈S(R)such ha  −ϕL1(ν)<ε 4. Then we ha e  χEnL1(ν)≤2 χEn−ϕχEnL1(ν)+2ϕχEnL1(ν) ≤2 −ϕL1(ν)+2ϕχEnL1(ν)<ε 2+2ϕχEnL1(ν) and knowing ha ϕχEnL1(ν)→0, i ollows ha  χEnL1(ν)→0. 2) =⇒3) Le 0 ≤ n↑ ∈L1(ν)and le ε >0. The Lemma 3.3 assu es ha he e exis s B∈Rloc, wi h 0 <ν(B) <∞(we assume ha is no he null unc ion), such ha  χΩBL1(ν)<ε 24 . Fo e e y n ≥1 conside he measu able subse s En:=  − n>ε 12ν(B)∈Rloc. No e ha En↓∅. By he hypo hesis  χEnL1(ν)<ε 24 o la ge enough n. Then o hose nwe ha e ha  − nL1(ν)≤2( − n)χΩBL1(ν)+2( − n)χBL1(ν) ≤4 χΩBL1(ν)+4 nχΩBL1(ν) +4( − n)χEnL1(ν)+4( − n)χBEnL1(ν) ≤8 χΩBL1(ν)+8 χEnL1(ν) +4ε 12ν(B)ν(BEn)<8ε 24 +8ε 24 +4ε 12 =ε, and  − nL1(ν)→0. 3) =⇒1) Le (An)n⊆Rbe a disjoin sequence wi h ν (∪n≥1An)<∞. Pu Bn:= A1∪···∪An o e e y n ≥1. Then 0 ≤χBn↑χA, whe e A := ∪n≥1An, since he sequence (An)nis pai wise disjoin . Mo eo e χA∈L1(ν), as ν(A) <∞. By he hypo hesis i ollows ha χA−χBnL1(ν)→0, bu ν(An+1)X≤ν(An+1)≤ν(Bn+1)=χBn+1 L1(ν)≤χA−χBnL1(ν). 3) ⇐⇒ 4) This equi alence ollows om he defini ion o he space Lp(ν)and he ac ha i is no mable as we ha e commen ed p e iously. 2 Rema k 3.7. Now, knowing ha Lp(ν)has o de con inuous no m i he measu e νis locally s ongly addi i e, i is no difficul o see ha S(R)is dense in Lp(ν) o e e y 1 ≤p <∞. 4. Reflexi i y o Lpo he semi a ia ion Example 3.2 ells us ha no always Lp(ν)is a eflexi e space e en o p >1. In his sec ion we cha ac e ize hose ec o measu es ν:R →Xsuch ha Lp(ν)is eflexi e. Fi s we need he ollowing echnical esul s which a e in e es ing in hemsel es. P oposi ion 4.1. Fo e e y p >1, he space Lp(ν)is a -con ex Banach la ice o e e y 1 ≤ <p. P oo . As commen ed abo e, we know ha Ls(ν)is a Banach la ice o he equi alen la ice no m  ·s whene e s >1. In o de o p o e ha Lp(ν)is -con ex i is enough o show ha he e exis s K>0 such ha   (| 1| +···+| n| )1   Lp(ν)≤K 1 Lp(ν)+···+ n Lp(ν)1 , o e e y elec ion o ec o s 1, ..., nin Lp(ν). Take in o accoun ha s := p >1, and so he e exis wo cons an s C1, C2>0such ha C1hLs(ν)≤hs≤C2hLs(ν),h∈Ls(ν). Recall also ha  Lp(ν)=| | 1 Ls(ν) o all ∈Lp(ν)o , equi alen ly,   |h|1   Lp(ν)=h1 Ls(ν) o all h ∈Ls(ν). Then, o e e y elec ion o ec o s 1, ..., nin Lp(ν), we ha e      n  k=1 | k| 1      Lp(ν) =     n  k=1 | k|      1 Ls(ν) ≤1 C1     n  k=1 | k|      1 s ≤1 C1n  k=1 | k| s1 ≤C 1 2 C1n  k=1 | k| Ls(ν)1 ≤C 1 2 C1n  k=1  k Lp(ν)1 as we wan o p o e. 2 P oposi ion 4.2. Le ν:R →Xbe a ec o measu e. Fo e e y 1 <p <∞, he inclusions L1 w(ν) ∩L∞(ν) ⊆ Lp(ν) ⊆L1 w(ν) +L∞(ν)hold. P oo . Fo he second inclusion no e ha Lp(ν) ⊆Lp w(ν). Now, i ∈Lp w(ν) decompose i as = χ[| |>1] + χ[| |≤1]. I is clea ha χ[| |≤1] ∈L∞(ν). On he o he hand, o p ≥1, we ha e | |χ[| |>1] ≤| |pχ[| |>1] ≤| |p∈L1 w(ν), and | |χ[| |>1] ∈L1 w(ν). Consequen ly ∈L1 w(ν) +L∞(ν).